Straight Disclination Research Articles

Structure of twin boundaries of cholesteric blue phase I

We investigate numerically the structure of twin boundaries of cholesteric blue phases. Our study is based on the Landau-de Gennes continuum theory describing the orientational order of the liquid crystal by a second-rank tensor. We pay particular attention to blue phase I (BP I) with body-centered-cubic symmetry and consider twin boundaries between BP I lattices in which their (110) planes are shared and the (1[over ¯]12) plane of one lattice is parallel to the (11[over ¯]2) plane of the other as observed in previous experiments [Jin etal., Sci. Adv. 6, eaay5986 (2020)10.1126/sciadv.aay5986; Zhang etal., ACS Appl. Mater. Interf. 13, 36130 (2021)1944-824410.1021/acsami.1c06873]. We discuss two plausible cases in which the twin boundaries are parallel to the {112} planes or the {111} planes. In the former, disclination lines of obtusely bent form penetrate the twin boundaries, and in the latter straight disclination lines as well as bent ones are found at the twin boundaries. The former twin boundaries are energetically less costly, consistent with previous experimental identifications. From our numerical results the free energy of a twin boundary per unit area is estimated to be ≃4×10^{-6}Jm^{-2}, which indeed indicates that the formation of twin boundaries is not prohibitively costly.

Straight and curved disclinations and dislocation equivalents

ABSTRACT Traditionally, for hard materials, two types of straight disclinations are identified although some Somigliana disclinations with different aspects have been treated theoretically. Here, we consider four physically distinct types of curved generalised disclinations, one of which reduces to two straight types, giving a total of five. We indicate typical applications for the different types.

A revisit of the elastic fields of straight disclinations with new solutions for a rigid core

The classical solutions for straight disclinations in an infinite elastic solid have been obtained by integrating the results for disclination densities. In this paper, the equilibrium equations are solved directly for straight twist and wedge disclinations, subject to the boundary conditions of the defects and rigid body translations/rotations. For a twist or wedge disclination in an infinite solid, the current solutions, based on a core fixed at a point to remove rigid body motion, differ from the classical ones by the constant $$-\hbox {log }r_{i}$$ , where $$r_{i }$$ is the radius of the disclination core. For a wedge disclination in an infinitely long cylinder, additional terms of the form 1 / r in the radial displacement and $$1/r^{2}$$ in the stresses appear in the solutions. The dependence of the current and classical results on the Lame constants highlights significant differences near the disclination line, which will impact studies of disclination relaxation such as crack nucleation and core amorphization. The energy of a singular wedge disclination in a cylinder without a core mostly underestimates that of a wedge disclination with a core.

Description of Disclinations and Dislocations by the Chern–Simons Action for $$\mathbb{S}\mathbb{O}(3)$$ Connection

We obtained the exact solution of the Euler–Lagrange equations following from the Chern–Simons action for $$\mathbb{S}\mathbb{O}(3)$$ connection with δ-type source. This solution is proved to describe straight linear disclination in the framework of geometric theory of defects. Torsion tensor components are calculated assuming the metric to be Euclidean. It shows that disclination can be followed by continuous distribution of dislocations with cylindrical symmetry.

Twist-induced crossover from two-dimensional to three-dimensional turbulence in active nematics.

While studies of active nematics in two dimensions have shed light on various aspects of the flow regimes and topology of active matter, three-dimensional properties of topological defects and chaotic flows remain unexplored. By confining a film of active nematics between two parallel plates, we use continuum simulations and analytical arguments to demonstrate that the crossover from quasi-two-dimensional (quasi-2D) to three-dimensional (3D) chaotic flows is controlled by the morphology of the disclination lines. For small plate separations, the active nematic behaves as a quasi-2D material, with straight topological disclination lines spanning the height of the channel and exhibiting effectively 2D active turbulence. Upon increasing channel height, we find a crossover to 3D chaotic flows due to the contortion of disclinations above a critical activity. Above this critical activity highly contorted disclination lines and disclination loops are formed. We further show that these contortions are engendered by twist perturbations producing a sharp change in the curvature of disclinations.

Chern—Simons Action and Disclinations

We review the main properties of the Chern—Simons and Hilbert—Einstein actions on a three-dimensional manifold with Riemannian metric and torsion. We show a connection between these actions that is based on the gauge model for the inhomogeneous rotation group. The exact solution of the Euler—Lagrange equations is found for the Chern—Simons action with the linear source. This solution is proved to describe one straight linear disclination in the geometric theory of defects.

Chern-Simons term in the geometric theory of defects

The Chern--Simons term is used in the geometric theory of defects. The equilibrium equations with $\delta$-function source are explicitly solved with respect to the $SO(3)$ connection. This solution describes one straight linear disclination and corresponds to the new kind of geometrical defect: it is the defect in the connection but not the metric which is the flat Euclidean metric. This is the first example of a disclination described within the geometric theory of defects. The corresponding angular rotation field is computed.

Nematic liquid crystals on sinusoidal channels: the zigzag instability

Substrates which are chemically or topographically patterned induce a variety of liquid crystal textures. The response of the liquid crystal to competing surface orientations, typical of patterned substrates, is determined by the anisotropy of the elastic constants and the interplay of the relevant lengths scales, such as the correlation length and the surface geometrical parameters. Transitions between different textures, usually with different symmetries, may occur under a wide range of conditions. We use the Landau–de Gennes free energy to investigate the texture of nematics in sinusoidal channels with parallel anchoring bounded by nematic-air interfaces that favour perpendicular (hometropic) anchoring. In micron size channels 5CB was observed to exhibit a non-trivial texture characterized by a disclination line, within the channel, which is broken into a zigzag pattern. Our calculations reveal that when the elastic anisotropy of the nematic does not favour twist distortions the defect is a straight disclination line that runs along the channel, which breaks into a zigzag pattern with a characteristic period, when the twist elastic constant becomes sufficiently small when compared to the splay and bend constants. The transition occurs through a twist instability that drives the defect line to rotate from its original position. The interplay between the energetically favourable twist distortions that induce the defect rotation and the liquid crystal anchoring at the surfaces leads to the zigzag pattern. We investigate in detail the dependence of the periodicity of the zigzag pattern on the geometrical parameters of the sinusoidal channels, which in line with the experimental results is found to be non-linear.

Surface-frustrated periodic textures of smectic-Aliquid crystals on crystalline surfaces

Using polarizing optical microscopy we studied thin films and droplets of smectic- A 4-cyano- 4;{'} -n-octylbiphenyl (8CB) liquid crystal deposited in air on crystalline surfaces of muscovite mica that induce monostable planar anchoring. The competition with the homeotropic anchoring at the 8CB-air interface leads to the formation of one-dimensional (1D) patterns composed of straight, parallel defect domains that are organized in periodic arrays over areas as large as several mm;{2} . We have developed a simple model which identifies the arrays with self-assembled "oily streaks," comprising straight disclination lines and curvature walls. The model reproduces the observed monotonic increase of the period p with the film thickness h in the range p=1-4 microm and h=0.8-17 microm. For higher values of h we observed a sharp transition to a 2D lattice of fragmented focal conic domains. Despite the apparent generality of our model for hybrid planar-homeotropic anchoring conditions, periodic arrays of straight oily streaks have been observed so far only for 8CB on crystalline surfaces such as mica or MoS2 . Our model indicates that this specificity is due to a particularly strong anchoring of the liquid crystal on such surfaces.

Wedge and twist disclinations in second strain gradient elasticity

The aim of this paper is to study disclinations in the framework of a second strain gradient elasticity theory. This second strain gradient elasticity has been proposed based on the first and second gradients of the strain tensor by Lazar et al. [Lazar, M., Maugin, G.A., Aifantis, E.C., 2006. Dislocations in second strain gradient elasticity. Int. J. Solids Struct. 43, 1787–1817]. Such a theory is an extension of the first strain gradient elasticity [Lazar, M., Maugin, G.A., 2005. Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity. Int. J. Eng. Sci. 43, 1157–1184] with triple stress. By means of the stress function method, the exact analytical solutions for stress and strain fields of straight disclinations in an infinitely extended linear isotropic medium have been found. An important result is that the force stress, double stress and triple stress produced by wedge and twist disclinations are nonsingular. Meanwhile, the corresponding elastic strain and its gradients are also nonsingular. Analytical results indicate that the second strain gradient theory has the capacity of eliminating all unphysical singularities of physical fields.

Fields of stress and electric displacement produced by dislocations and disclinations in three-dimensional, anisotropic, elastic, and piezoelectric media. II. Infinite straight dislocation and Frank disclination

The fields of stress and electric displacement caused by infinitely extended straight dislocations and Frank disclinations are deduced from the author's statements for the fields caused by a continuous distribution of dislocations and disclinations (S. Minagawa, Phil. Mag. 84 2229 (2004)). The multiple integrals in the original statements are converted into functions of space coordinates. Cauchy's theorem plays an important part. The improper integral that appears in computations of the fields around a Frank disclination is interpreted as its finite part by Hadamard. Examples are the fields around an infinite straight defect in caesium copper chloride, as well as those in gallium arsenide. The contours and zero lines are plotted to illustrate the fields caused by a dislocation and a disclination dipole.

Instability modes of high-strength disclinations in nematics

We solve the complete tensor fluctuation problem of a long and straight nematic disclination line with a general winding number in the one elastic constant approximation. Focusing on the eigenmodes growing in time, we show that the disclination with strength higher than 1/2 is unstable with respect to the splitting and for integer strength also to the escape--in both cases there is no metastability. Numerically we show that a moderate elastic anisotropy, e.g., as found in thermotropic liquid crystals like 5CB or MBBA, does not introduce any metastability either.

Twist disclination in the field theory of elastoplasticity

In this paper we study the twist disclination within the elastoplastic defect theory. Usingthe stress function method, we found exact analytical solutions for all characteristic fieldsof a straight twist disclination in an infinitely extended linear isotropic medium. The elasticstress, elastic strain and displacement have no singularities at the disclination line. Wefound modified stress functions for the twist disclination. In addition, we calculate thedisclination density, effective Frank vector, disclination torsion and effective Burgers vectorof a straight twist disclination. By means of gauge theory of defects we decompose theelastic distortion into the translational and rotational gauge fields of the straight twistdisclination.

Dislocation and disclination loops in the virtual-defect method

A method of virtual circular defect loops is developed for determining the elastic fields produced by defects in a bounded medium in the case of an axially symmetric geometry. In this method, continuously distributed virtual circular Volterra and Somigliana dislocation loops are adjusted in such a way as to satisfy the boundary conditions imposed at free surfaces and interfaces. Original calculations of the elastic fields of circular defect loops of different types are carried out. The elastic fields are found for the case of straight dislocations and disclinations in a plate that are perpendicular to the plate plane and for the case of circular disclination loops parallel to the plate plane or to an interface.

Dislocations and disclinations in the gradient theory of elasticity

The results of application of gradient theory of elasticity to a description of elastic fields and dislocation and disclination energies are considered. The main achievement made in this approach is the removal of the classical singularities at defect lines and the possibility of describing short-range interactions between them on a nanoscopic level. Non-singular solutions for stress and strain fields of straight disclination dipoles in an infinite isotropic medium have been obtained within a version of the gradient theory of elasticity. A description is given of elastic fields near disclination lines and of specific features in the short-range interactions between disclinations, whose study is impossible to make in terms of the classical linear theory of elasticity. The strains and stresses at disclination lines are shown to depend strongly on the dipole arm d. For short-range interaction between disclinations, where d varies from zero to a few atomic spacings, these quantities vary monotonically for wedge disclinations and nonmonotonically in the case of twist disclinations, and tend uniformly to zero as disclinations annihilate. At distances from disclination lines above a few atomic spacings, the gradient and classical solutions coincide. As in the classical theory of elasticity, the gradient solution for the wedge-disclination dipole transforms to the well-known gradient solution for a wedge dislocation at distances d substantially smaller than the interatomic spacing.

Dislocations and Disclinations in Gradient Elasticity

A special gradient theory of elasticity is employed to consider dislocations and disclinations with emphasis on the elimination of strain singularities appearing in the classical theory of elasticity. For dislocations, we give a brief summary of our earlier results pertaining to “non-singular” expressions for the elastic strains, as well as new results for “non-singular” expressions for the strain energies. For disclinations, we derive non-singular expressions for the elastic strains demonstrating that dipoles of straight disclinations of general type give zero or finite values for the strain components at the disclination line. The finite values depend strongly on the dipole arm d and exhibit a regular monotonous (wedge disclinations) or non-monotonous (twist disclinations) behavior for short-range (d 〈 10 √c) interactions. At annihilation distances (d → 0), the elastic strains tend smoothly to zero. Far from the disclination line (r 〉〉 √c), gradient and classical solutions coincide. When the dipole arm d is much smaller than the scale unit √c, the elastic fields of a dipole of wedge disclinations transform into the elastic fields of an edge dislocation, as is the case in classical elasticity.

Straight disclinations in nonlocal elasticity

The solution for the problems of straight disclinations with various directions of the Frank vector in an infinitely extended linear isotropic nonlocal elastic medium are derived assuming nonlocal moduli of two types. It is found that the stress fields have no singularities at the disclination lines but the logarithmical singularity at infinity remains as in the classical solution. Such a singularity is removed in a finite body, for example, in a case of circular cylinder with radius R.

Dislocations and disclinations in transversely isotropic elastic solids

Abstract A recently developed method to obtain the elastic fields due to defects in isotropic solids has been extended to include the case of fields due to dislocations and disclinations with arbitrary orientations in transversely isotropic infinite solids. The extension hinges on the introduction of a newly defined stress vector function, the hexagonal stress vector, which, together with the Green functions for double force and for double force with moment, yields closed-form solutions. The solution is first obtained for infinitesimal Volterra dislocation loops. The elastic fields due to dislocations and disclinations are then obtained by integrating the solution for the infinitesimal loop over the surface of the cut that generates the line defects. Along with the Newtonian and biharmonic potential functions that are used for isotropic solids, two more harmonic potential functions are introduced to express the solution in transversely isotropic solids. Closed-form solutions for straight dislocation and disclination lines, and circular dislocation loops are presented as examples to illustrate the application of the method.

Unpinning of 2π-twist disclination from impurities in chiral smectic C liquid crystals

Point-like impurities in chiral smectic C liquid crystals modelled by distributions of infinitesimal twist disclination loops can pin straight 2π-twist disclinations. Under the influence of the helicoidal pitch temperature change or by the application of an external electric field the unpinning of a disclination occurs. Approximate description of the unpinning process shows that the importance of the disclination pinning decreases with the sample thickness.

Twist disclination loops in chiral smectic C liquid crystals

The notion of an infinitesimal twist disclination loop known from solids is introduced into a chiral smectic C liquid crystal with parallel layers. Finite rectangular and straight twist disclinations are described as a synthesis of a planar distribution of infinitesimal loops. Then a small impurity which disturbs only the molecular arrangement in one smectic layer is modelled by a pair of infinitesimal twist disclination loops. The interaction between a 2π-twist disclination and a small impurity is estimated using the interaction energy between a straight 2π-twist disclination and an infinitesimal twist disclination loop.